# Combined Rates Quant Question

by on October 9th, 2010

Combined Rates questions can be daunting for anyone taking the GMAT.

Let’s take a look at this sample question and detailed solution.  Challenge yourself by trying out the problem first before reviewing the solution.

Car X began traveling at an average speed of 35 miles per hour.  After 72 minutes, car Y began traveling at an average speed of 49 miles per hour.  When both cars had traveled the same distance, both cars stopped.  How many miles did car X travel from the time car Y began traveling until both cars stopped?

(A) 105
(B) 120
(C) 140
(D) 147
(E) 168

## Solution:

Step 1 is to determine how far car X traveled during the 72 minutes before car Y has started.  To do this, we first convert 72 minutes to 6/5 of an hour (that’s 72/60 simplified), in order to make sure our units match, as our speed is given in miles per hour.  Since rate * time = distance, we know that car X traveled 35mph * 6/5 hours = 42 miles.

Step 2 we need to determine a relative rate. We do this in order to figure out how many miles car Y gains on car X each hour.  To calculate this, subtract the rate of car X from the rate of car Y.  49mph – 35mph = 14mph, which is our relative rate in miles per hour, or the rate that Y gains on X per hour.

Step 3 take the distance car Y must gain on car X (that’s 42 miles, from step 1) and divide it by the number of miles car Y gains on X in an hour (that’s 14 mph from step 2) in order to find the number of hours it will take for both cars to travel the same distance: 42mi/14mph = 3 hours.

Finally, we are asked to find how far car X traveled after car Y starts.  We use the time we calculated from step 3, which is 3 hours, and multiply it by car X’s rate (35mph).  This gives us 3hrs * 35mph = 105 miles, which is choice (A) and the correct answer to this problem.

• Too easy !!

Did this one completely mentally and in just a little over 2 mins...Actual GMAT questions are a lot lot harder than above. Certainly not 'Advanced Math' level as per current GMAT trends.

• Hey Bret,
Good Question. While i was solving this i used a little different approach after step 1.
Once we know that X has traveled 42 miles after which Y starts, we can assume that X travels a further distance of 'a' miles such that the distance traveled by Y and X are same.
Now the time taken for X to travel 'a' and for Y to travel '42+a' are the same hence we can equate as follows:-
(42 + a) / 49 = a / 35

solving for a we will get a = 105.
This takes less than 30 secs to solve if we know the formula Distance = speed * time

Ishaan

• Excellent Ishaan...Thanks for sharing your logic...

• *confused* *help!*

"When both cars had traveled the same distance, both cars stopped."

How does this mean they both stopped @ once?
If car A & car B leave out at once... car A goes @ 100 kmph and stops after an hour (100 km) .. and car B goes @ 1 kmph and stops after 100 hours (100 km) ...

Is it coz it says "when....had..." ? wud it be different if it said "Both cars stopped after travelling the same distance" ??? .. watever tht distance might be...

• Hey rohit,

Its simple. Try to understand the concept this way.
Assume you and your kid brother are running a race, now since your brother is small, you will give him a headstart (I hope you will ). After sometime you will start running yourself.
Since you are faster than your brother, soon you will catch up to him. At that moment you stop the race. This way both of you have traveled the same distance.
the total time taken by your brother is actually the time taken by you as well as the time of headstart you gave him.

As per your query, car A and B start at the same time. But in the question, X has a headstart.
I hope you get it.
Ishaan